A general eciency relation for molecular machines Milo M. Lin Green Center for Systems Biology Department of Bioinformatics Department of Biophysics and the Center for

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A general eﬃciency relation for molecular machines

Milo M. Lin

Green Center for Systems Biology, Department of Bioinformatics, Department of Biophysics, and the Center for

Alzheimer’s and Neurodegenerative Diseases, University of Texas Southwestern Medical Center, Dallas, TX 75390

Milo.Lin@UTSouthwestern.edu

Living systems eﬃciently use chemical fuel to do work, process information, and assemble patterns

despite thermal noise. Whether high eﬃciency arises from general principles or speciﬁc ﬁne-

tuning is unknown. Here, applying a recent mapping from nonequilibrium systems to battery-

resistor circuits, I derive an analytic expression for the eﬃciency of any dissipative molecular

machine driven by one or a series of chemical potential diﬀerences. This expression disentangles

the chemical potential from the machine’s details, whose eﬀect on the eﬃciency is fully speciﬁed by

a constant called the load resistance. The eﬃciency passes through a switch-like inﬂection point

if the balance between chemical potential and load resistance exceeds thermal noise. Therefore,

dissipative chemical engines qualitatively diﬀer from heat engines, which lack threshold behavior.

This explains all-or-none dynein stepping with increasing ATP concentration observed in single-

molecule experiments. These results indicate that biomolecular energy transduction is eﬃcient

not because of idosyncratic optimization of the biomolecules themselves, but rather because the

concentration of chemical fuel is kept above a threshold level within cells.

Energy is a limiting resource for life. Therefore, molecular machines that harness diﬀerences in chemical

potential ∆µto perform tasks within the cell must function as highly eﬃcient chemical engines. Despite

thermal stochasticity, the measured eﬃciencies of biomolecular chemical engines are consistently in the 60-to-

90 percent range (1–3), well above that of typical human-designed systems such as heat engines. Two centuries

ago, Carnot showed that the maximum eﬃciency of heat engines is (4):

ηheat ≤∆T

∆T+T0

.(1)

This relation depends on two independent parameters: the temperature diﬀerential driving the engine ∆Tand

the exhaust temperature T0. Eq. 1 provides a universal constraint for all heat engines regardless of design

speciﬁcs, and initiated the ﬁeld of thermodynamics. Finding a general relation constraining the behavior

of chemical engines would provide a unifying framework for biomolecular function and evolution, and for

engineering nanodevices for physiological conditions.

It is important to distinguish two broad classes of chemical engines in biology, as they should have diﬀerent

formulations of eﬃciency as well as design constraints. The ﬁrst is the class of energy transduction machines,

such as ion pumps, that convert energy from one chemical reservoir to another. In these cases, it is clear that

any energy dissipated by the machine is wasted and so should be minimized in order to maximize the eﬃciency

of transferring energy between the reservoirs, which is achieved in the ”tight-coupling” limit (5).

This work is primarily concerned with the second class of chemical engines whose purpose is the dissipation

of the input energy. This class of dissipative machines serve a wide range of functions, including mechanical

transport against viscous drag, and maintaining patterns of protein assembly or signaling conﬁgurations that

are inaccessible at equilibrium (Fig. 1A). They are therefore principal agents of information processing and

structural reorganization within the cell. Prigogine called these conﬁgurations ”dissipative structures” because

input energy is constantly dissipated to the environment as heat to maintain these patterns, which are far

from equilibrium even under steady state conditions (6). Chemical engines can approach perfect eﬃciency if

the throughput and power approach zero (5; 7). Therefore, unlike Carnot’s relation Eq.1, a useful general

relation for the eﬃciency of chemical engines must depend on at least one system-dependent collective variable

that reﬂects the constraint on throughput in the high eﬃciency limit, and reveal if there are regimes of the

collective variable and ∆µfor which the machine is both fast and eﬃcient. Therefore, such a relation could

also reveal if achieving high performance in both metrics requires evolutionary ﬁne-tuning, or is a general

property. We currently lack mechanistic insight into how chemical engine eﬃciency is controlled by tunable

parameters. And it is unclear if the multitude of system-speciﬁc parameters could be encapsulated by a single

collective variable that is well deﬁned for all systems, which is necessary for a clear uniﬁed understanding.

Existing results on dissipative chemical engines, based on generalized ﬂuctuation-dissipation relations

(8; 9), upper-bound the eﬃciency relative to observed ﬂuctuations (10; 11), although the bound may not

be tight (achievable) far from equilibrium. Existing work does provide eﬃciency constraints under limiting

conditions. For example, weakly driven engines operating at maximum power must be 50% eﬃcient (12–

14). However, this bound only holds near equilibrium, and does not apply to the strongly-driven conditions

relevant for biology. Numerical simulations of simple chemical engines indicate that eﬃciency is increased

when driven farther from equilibrium by increasing ∆µ, similar to the eﬀect of increasing ∆Tin heat engines

(12). This is consistent with a model in which molecular motors can become less wasteful if driven by higher

arXiv:2210.04380v1 [physics.bio-ph] 10 Oct 2022

FIG. 1 Mapping nonequilibrium cycles to battery-resistor circuits. Cellular function and homeostasis are maintained by

cyclic molecular engines that harness chemical energy to transform and organize matter (simpliﬁed schematic of a cell in A;

energy ﬂow in red). The circuit mapping transforms a dynamical system, for example an enzymatic engine that activates a

signaling molecule using energy released from hydrolyzing ATP into ADP and phosphate (B; star denotes activated enzyme

complex), into an electronic circuit consisting of batteries and resistors that obey Ohm’s law (C). Subsequently, the circuit

elements can be systematically simpliﬁed to describe coarse-grained probabilities and currents without loss of accuracy (D).

ATP concentration (15), and suggest a general link between large ∆µand high eﬃciency.

Here, using a recent generalization of the Boltzmann distribution to nonequilibrium systems(16), I derive

a general equation for the maximum eﬃciency of dissipative chemical engines. This relation depends on

two independent tunable variables: ∆µand the relative load resistance. The load resistance is a collective

variable condensing the engine details, and is a constant unaﬀected by how strongly the engine is driven (i.e.

∆µ). The equation shows generally that biomolecular processes are intrinsically eﬃcient because ∆µis much

larger than the thermal energy kT in vivo. However, unlike heat engines, chemical engines have an eﬃciency

inﬂection point at a threshold ∆µ/kT that depends on the load resistance. This switch-like transition explains

single-molecule in vitro measurements of dynein molecular motor stepping. This result also leads to a general

tradeoﬀ relation between energy eﬃciency and time eﬃciency. More broadly, the existence of a general

eﬃciency inﬂection point provides a candidate thermodynamic necessary condition for life.

Circuit mapping. Chemical engines within the cell (circular arrows in Fig. 1A) consume energy to

form patterns of activated signaling molecules, protein self-assembly, or directed movement, that would

be vanishingly rare at equilibrium. Such processes can be modeled as nonequilibrium Markov chains, but

obtaining analytical insights into such systems has mostly been intractable (17–19). This work exploits a

recently found mapping from the dynamical network of any Markovian system to an electronic circuit (16)

(See, for example, Fig. 1C,D). Consequently, a system is decomposed into a passive equilibrium portion

(”resistors”) that is driven by chemical potential diﬀerences ∆µ(”batteries”). The resistance between two

neighboring states mand nis: Rmn ≡eβGm

kmn , where β=1

kT ,Gmis the free energy of state mat equilibrium

(i.e. in the absence of driving), and kmn is the equilibrium rate constant of transitioning from state mto n. If

a transition is directly driven by ∆µ, this transition is mapped to a battery with a probability potential drop

Emn ≡(eβ∆µ−1)eβGmPm(red arrow in Fig. 1B). Using this mapping, probability ﬂow amongst the states

of any system obeys Ohm’s law (16): PjeβGj−PieβGi=

n=j

m=i

(Emn −RmnImn) (16), where the summation

is along any trajectory (parameterized by neighboring states m, n) connecting states iand j.Imn is the

probability current (net ﬂux) from state mto state n. Note that Emn and Imn are zero for a system at

equilibrium, and Ohm’s law reduces to the Boltzmann distribution Pj=Pieβ(Gi−Gj)in this case, as expected.

In this mapping, multiple interconnected dynamical transitions can be systematically combined into a single

resistor, called the Thevenin equivalent resistance (20), that captures their collective eﬀect on the probability

ﬂux at steady state. For example, for the signaling protein in Fig.1B, molecular transitions corresponding

to R1, .., R6(Fig. 1C) can be combined as series and parallel resistors into the equivalent resistance

RL=R1+ ( 1

R2+1

R3+R4+R5+R6)−1(Fig. 1D). Resistor coarse-graining, combined with the broad set of circuit

theorems, allows a systematic approach to simplify systems of otherwise intractable complexity (16). This ap-

proach is most powerful for systems, such as biomolecular machines, for which the driven transitions are sparse.

Deﬁning the eﬃciency of dissipative chemical engines. Dissipative machines within the cell are

usually composed of proteins that occupy one of many possible conformation and interaction states (Fig. 1B).

Each machine harnesses a source of chemical energy, often by binding excess ATP and coupling its hydrolysis

into ADP and phosphate to drive the target reaction. Consider the circuit representation of a machine driven

by the source transition S (i.e. battery) with ∆µ=kT ln 1 + αS

kS, where kSis the apparent rate constant of

the source transition in the absence of driving (e.g. no excess ATP at equilibrium), and αSis the enhancement

in the rate constant due to driving. If the energy source is ATP, the source transition is enhancement of

the rate of converting unbound phosphate and ADP into ATP bound to the protein (red arrow in Fig. 1B).

At ATP concentrations well below binding saturation, αSis proportional to ATP concentration in excess

of the equilibrium concentration. The resistance of the source transition (i.e. battery) is denoted RS(Fig.

2A). A machine may contain multiple source transitions. Although the circuit framework is compatible with

arbitrary arrangements of batteries, the analytic results in this work are restricted to those machines with a

single source transition or multiple source transitions in series. Nevertheless, this class of machines includes a

large fraction of cellular processes including signaling, self-assembly and sorting, and mechanical transport.

The source transition breaks detailed balance for the machine by driving net ﬂux of downstream transitions

between states, and this ﬂux may take numerous alternate pathways through state space. All transitions aside

from the source transition can be combined into a single Thevenin equivalent load resistor RL(Fig. 2A).

Often, the biologically useful output of the engine is to enhance a particular state within the load.

With this formulation, we can now deﬁne the maximum eﬃciency of a dissipative machine as the

dissipation rate of the load divided by the total dissipation rate. This deﬁnition corresponds to existing

formulations of eﬃciency as special cases. For example, in the case that the machine is a molecular motor

transporting a cargo against viscous drag, the load dissipation coincides with the maximum mechanical work

that can be done by the motor per cycle. In this case, the eﬃciency is equivalent to the Stokes eﬃciency,

which is the average viscous drag times the average velocity divided by the total dissipation rate (21). More

broadly, this deﬁnition also extends the concept of eﬃciency to most processes in the cell, such as signaling

and pattern formation, in which the useful dissipation of energy does not correspond to work along a simple

reaction coordinate.

A general eﬃciency relation for dissipative chemical engines. We ﬁrst obtain the total dissi-

pation rate σT , where σis the entropy production rate. The nonequilibrium ﬂuctuation theorems (22–24)

relate dissipation rate to transition probabilities: σT =kT Pij Iij ln "Pi(kij +αij )

Pjkji #(18). Because dissipation

is summed over all possible dynamical transitions of a system, it may appear that knowing an engine’s details

is required to calculate the eﬃciency. Instead, I show here that RLis the only information about the load

needed to calculate the dissipation, and therefore the eﬃciency. First, using Tellegen’s Circuit Theorem (25),

the total steady-state dissipation rate is (See Supp. Information):

σT =X

Iij ∆µij .(2)

In this reformulation, the total dissipation rate is a function of only the currents of the directly driven transitions

(for which αij 6= 0), even though energy is dissipated at all transitions. Eq. 2 is especially useful if driven

transitions are sparse.

If there is a single driven transition, the steady-state load dissipation rate is (Supp. Information): σLT=

kT ISln RS+RLe∆µ/kT

RS+RL. The diﬀerence between σT and σLTis the dissipation of the source (battery), which

corresponds to energy that is not accessible to the load. The eﬃciency is upper-bounded because of the

dissipation of the source, which corresponds to the energy lost whenever the engine dynamics happens to

reverse the source transition, for example, if bound ATP is transformed into unbound ADP and phosphate

without coupling to any downstream load reactions. The maximum steady state eﬃciency is: ηchem ≤σL/σ.

This yields the main result of this work, a general relation governing the maximum eﬃciency of dissipative

chemical engines:

ηchem ≤kT

∆µln "RS

RL+e∆µ

RL+ 1 #.(3)

The eﬃciency depends on the chemical potential diﬀerence relative to kT , which quantiﬁes how the eﬃciency

is limited by thermal ﬂuctuations. The functional form of Eq. 3 remains valid in the case that the engine is

cumulatively powered by multiple driven transitions (e.g. sequential hydrolysis of multiple ATP molecules);

in this general case, ∆µis replaced by the sum of ∆µ’s and RSis replaced by the weighted sum of all driven

resistors (See Supp. Information). An engine will approach the maximum eﬃciency set by Eq. 3 if the most

dissipative reactions within the load are useful. In practice, the load may include nonproductive futile cycles

which dissipate energy subsequent to hydrolysis.

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AgeneraleciencyrelationformolecularmachinesMiloM.LinGreenCenterforSystemsBiology,DepartmentofBioinformatics,DepartmentofBiophysics,andtheCenterforAlzheimer'sandNeurodegenerativeDiseases,UniversityofTexasSouthwesternMedicalCenter,Dallas,TX75390Milo.Lin@UTSouthwestern.eduLivingsystemsecientlyusechem...

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A general eciency relation for molecular machines Milo M. Lin Green Center for Systems Biology Department of Bioinformatics Department of Biophysics and the Center for

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